Blogger have introduced new Newsreel (or News Bar) and Video Bar widgets you can add to your New Blogger blog if you have switched your template to Layouts, so that you can display the results of your chosen searches on Google News or YouTube and Google Video in your sidebar. The results are kept dynamically updated through the magic of AJAX.
It was possible to do this before (see this post) but the new widgets make it much easier for non-coders to include these widgets on their blogs via the Page Elements tab through "Add a Page Element":
The results on your blog would look something like this (I've shown the video bar and news bar side by side for convenience, but they'd probably be stacked vertically on top of each other in your sidebar - well you could add them to your footer etc if you prefer, see my page elements widgets code reference and tutorial for a page elements howto):
I've updated my page elements widgets code reference and tutorial to add info about these two new widgets, including the options you can set for these widgets, and the widget code which those page elements add to your New Blogger template in Edit HTML view - see that post for full info.
Basically at the moment you can only set the search terms, and (in the case of the newsreel) whether you want results to open in a separate window.
It only displays 4 results, in both those widgets. You can't at present change the number of results you want displayed on your blog.